Abstract

We consider the diffraction occurring when light is focused by a lens without spherical aberration through a planar interface between materials of mismatched refractive indices, which focusing produces spherical aberration. By means of a rigorous vectorial electromagnetic treatment that was previously developed for this problem [ Török et al., J. Opt. Soc. Am. A 12, 325 ( 1995)], the diffraction integrals are transformed into a form that is computable. Time-averaged electric energy density distributions in the region of the focused probe are numerically evaluated for air–glass and air–silicon interfaces as a function of lens numerical aperture and probe depth corresponding to a wide range of spherical aberration. Two-dimensional lateral (xy) and meridional (xz) electric energy density plots show how the energy, the size, and the position of the various axial and lateral maxima changed, providing new information concerning the above two important optical systems. The treatment also shows that the use of a lens without spherical aberration to focus into a second material is formally equivalent to the use of a lens with spherical aberration and a reduced solid semi-angle to focus into a single material.

© 1995 Optical Society of America

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References

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  1. P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
    [CrossRef]
  2. E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
    [CrossRef]
  3. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  4. A. Boivin, E. Wolf, “Electromagnetic field in the neighbourhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
    [CrossRef]
  5. A. Hardy, D. Treves, “Structure of the electromagnetic field near the focus of a stigmatic lens,” J. Opt. Soc. Am. 63, 85–90 (1973).
    [CrossRef]
  6. H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
    [CrossRef]
  7. R. Kant, “Analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
    [CrossRef]
  8. H. Ling, S-W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
    [CrossRef]
  9. J. J. Stamnes, Waves in Focal Regions, 1st ed. (Adam Hilger, Bristol, UK, 1986).
  10. P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation: errata,” J. Opt. Soc. Am. A 12, 1605 (1995).
    [CrossRef]
  11. It should be emphasized that this transformation is applicable only when the electric field is being computed inside the second material. When the diffraction integrals are applied to compute the field at the immediate vicinity of the interface and when n2> n1, the application of this transformation excludes the evanescent waves from the expression, and thus for the latter case the original equations should be used.
  12. M. R. Brozel, I. Grant, R. M. Ware, D. J. Stirland, “Direct observation of the principal deep level (EL2) in undoped semi-insulating GaAs,” Appl. Phys. Lett. 42, 610–612 (1983).
    [CrossRef]
  13. G. R. Booker, Z. Laczik, P. Kidd, “The scanning infrared microscope (SIRM) and its applications to bulk GaAs and Si: a review,” Semicond. Sci. Technol. 7, A110–A121 (1992).
    [CrossRef]
  14. P. Török, Z. Laczik, G. R. Booker, “The development and use of a new confocal SIRM incorporating reflection, double-pass and phase contrast modes,” Inst. Phys. Conf. Ser. 134, 771–774 (1993).

1995 (2)

1993 (2)

R. Kant, “Analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
[CrossRef]

P. Török, Z. Laczik, G. R. Booker, “The development and use of a new confocal SIRM incorporating reflection, double-pass and phase contrast modes,” Inst. Phys. Conf. Ser. 134, 771–774 (1993).

1992 (1)

G. R. Booker, Z. Laczik, P. Kidd, “The scanning infrared microscope (SIRM) and its applications to bulk GaAs and Si: a review,” Semicond. Sci. Technol. 7, A110–A121 (1992).
[CrossRef]

1984 (1)

1983 (1)

M. R. Brozel, I. Grant, R. M. Ware, D. J. Stirland, “Direct observation of the principal deep level (EL2) in undoped semi-insulating GaAs,” Appl. Phys. Lett. 42, 610–612 (1983).
[CrossRef]

1973 (1)

1970 (1)

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

1965 (1)

A. Boivin, E. Wolf, “Electromagnetic field in the neighbourhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Boivin, A.

A. Boivin, E. Wolf, “Electromagnetic field in the neighbourhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

Booker, G. R.

P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
[CrossRef]

P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation: errata,” J. Opt. Soc. Am. A 12, 1605 (1995).
[CrossRef]

P. Török, Z. Laczik, G. R. Booker, “The development and use of a new confocal SIRM incorporating reflection, double-pass and phase contrast modes,” Inst. Phys. Conf. Ser. 134, 771–774 (1993).

G. R. Booker, Z. Laczik, P. Kidd, “The scanning infrared microscope (SIRM) and its applications to bulk GaAs and Si: a review,” Semicond. Sci. Technol. 7, A110–A121 (1992).
[CrossRef]

Brozel, M. R.

M. R. Brozel, I. Grant, R. M. Ware, D. J. Stirland, “Direct observation of the principal deep level (EL2) in undoped semi-insulating GaAs,” Appl. Phys. Lett. 42, 610–612 (1983).
[CrossRef]

Grant, I.

M. R. Brozel, I. Grant, R. M. Ware, D. J. Stirland, “Direct observation of the principal deep level (EL2) in undoped semi-insulating GaAs,” Appl. Phys. Lett. 42, 610–612 (1983).
[CrossRef]

Hardy, A.

Hopkins, H. H.

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

Kant, R.

R. Kant, “Analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
[CrossRef]

Kidd, P.

G. R. Booker, Z. Laczik, P. Kidd, “The scanning infrared microscope (SIRM) and its applications to bulk GaAs and Si: a review,” Semicond. Sci. Technol. 7, A110–A121 (1992).
[CrossRef]

Laczik, Z.

P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
[CrossRef]

P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation: errata,” J. Opt. Soc. Am. A 12, 1605 (1995).
[CrossRef]

P. Török, Z. Laczik, G. R. Booker, “The development and use of a new confocal SIRM incorporating reflection, double-pass and phase contrast modes,” Inst. Phys. Conf. Ser. 134, 771–774 (1993).

G. R. Booker, Z. Laczik, P. Kidd, “The scanning infrared microscope (SIRM) and its applications to bulk GaAs and Si: a review,” Semicond. Sci. Technol. 7, A110–A121 (1992).
[CrossRef]

Lee, S-W.

Ling, H.

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions, 1st ed. (Adam Hilger, Bristol, UK, 1986).

Stirland, D. J.

M. R. Brozel, I. Grant, R. M. Ware, D. J. Stirland, “Direct observation of the principal deep level (EL2) in undoped semi-insulating GaAs,” Appl. Phys. Lett. 42, 610–612 (1983).
[CrossRef]

Török, P.

Treves, D.

Varga, P.

Ware, R. M.

M. R. Brozel, I. Grant, R. M. Ware, D. J. Stirland, “Direct observation of the principal deep level (EL2) in undoped semi-insulating GaAs,” Appl. Phys. Lett. 42, 610–612 (1983).
[CrossRef]

Wolf, E.

A. Boivin, E. Wolf, “Electromagnetic field in the neighbourhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

Yzuel, M. J.

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

Appl. Phys. Lett. (1)

M. R. Brozel, I. Grant, R. M. Ware, D. J. Stirland, “Direct observation of the principal deep level (EL2) in undoped semi-insulating GaAs,” Appl. Phys. Lett. 42, 610–612 (1983).
[CrossRef]

Inst. Phys. Conf. Ser. (1)

P. Török, Z. Laczik, G. R. Booker, “The development and use of a new confocal SIRM incorporating reflection, double-pass and phase contrast modes,” Inst. Phys. Conf. Ser. 134, 771–774 (1993).

J. Mod. Opt. (1)

R. Kant, “Analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Acta (1)

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

Phys. Rev. B (1)

A. Boivin, E. Wolf, “Electromagnetic field in the neighbourhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

Proc. R. Soc. London Ser. A (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Semicond. Sci. Technol. (1)

G. R. Booker, Z. Laczik, P. Kidd, “The scanning infrared microscope (SIRM) and its applications to bulk GaAs and Si: a review,” Semicond. Sci. Technol. 7, A110–A121 (1992).
[CrossRef]

Other (2)

It should be emphasized that this transformation is applicable only when the electric field is being computed inside the second material. When the diffraction integrals are applied to compute the field at the immediate vicinity of the interface and when n2> n1, the application of this transformation excludes the evanescent waves from the expression, and thus for the latter case the original equations should be used.

J. J. Stamnes, Waves in Focal Regions, 1st ed. (Adam Hilger, Bristol, UK, 1986).

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Figures (9)

Fig. 1
Fig. 1

Diagram showing light focused by a lens into two media separated by a planar interface.

Fig. 2
Fig. 2

Time-averaged electric energy density distributions in the xz meridional plane for a probe depth of 5 μm and for numerical apertures of (a) 0.3, (b) 0.6, and (c) 0.9. The calculations are for air (n1 = 1.0) and glass (n2 = 1.5) and a wavelength of λ = 0.6328 μm (He–Ne laser). 2.1E-3 is 2.1 × 10−3, and so on, in this and subsequent figures.

Fig. 3
Fig. 3

Time-averaged electric energy density distributions in the xz meridional plane for a numerical aperture of 0.9 and for probe depths of (a) 10, (b) 20, (c) 40, (d) 80, and (e) 160 μm. The calculations are for air (n1 = 1.0) and glass (n2 = 1.5) and a wavelength of λ = 0.6328 μm (He–Ne laser).

Fig. 4
Fig. 4

Axial scans of the time-averaged electric energy density distribution for numerical apertures of (a) 0.6 and (b) 0.9 as a function of probe depth from 0 to 280 μm. The calculations are for air (n1 = 1.0) and glass (n2 = 1.5) and a wavelength of λ = 0.6328 μm (He–Ne laser).

Fig. 5
Fig. 5

Time-averaged electric energy density distributions for a numerical aperture of 0.9 in the focal xy plane when the probe depth is 160 μm. The calculations are for air (n1 = 1.0) and glass (n2 = 1.5) and a wavelength of λ = 0.6328 μm (He–Ne laser).

Fig. 6
Fig. 6

Time-averaged electric energy density distributions in the xy meridional plane for a probe depth of 5 μm and for numerical apertures of (a) 0.3, (b) 0.6, and (c) 0.9. The calculations are for air (n1 = 1.0) and silicon (n2 = 3.5) and a wavelength of λ = 1.3 μm (infrared laser).

Fig. 7
Fig. 7

Time-averaged electric energy density distributions in the xz meridional plane for a numerical aperture of 0.9 and for probe depths of (a) 10, (b) 20, (c) 40, (d) 80, and (e) 160 μm. The calculations are for air (n1 = 1.0) and silicon (n2 = 3.5) and a wavelength of λ = 1.3 μm (infrared laser).

Fig. 8
Fig. 8

Axial scans of the time-averaged electric energy density distribution for numerical apertures of (a) 0.6 and (b) 0.9 as a function of probe depth from 0 to 280 μm. The calculations are for air (n1 = 1.0) and silicon (n2 = 3.5) and a wavelength of λ = 1.3 μm (infrared laser).

Fig. 9
Fig. 9

Time-averaged electric energy density distribution for a numerical aperture of 0.9 in the focal (xy) plane when the probe depth is 160 μm. The calculations are for air (n1 = 1.0) and silicon (n2 = 3.5) and a wavelength of λ = 1.3 μm (infrared laser).

Equations (11)

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s ^ 1 = ( sin ϕ 1 cos θ ) i ^ + ( sin ϕ 1 sin θ ) j ^ + ( cos ϕ 1 ) k ^ ,
s ^ 2 = ( sin ϕ 2 cos θ ) i ^ + ( sin ϕ 2 sin θ ) j ^ + ( cos ϕ 2 ) k ^ ,
r p = r p [ ( sin ϕ p cos θ p ) i ^ + ( sin ϕ p sin θ p ) j ^ + ( cos ϕ p ) k ^ ] ,
E ˜ ( P , t ) = Re [ E ( P ) exp ( - i ω t ) ] , H ˜ ( P , t ) = Re [ H ( P ) exp ( - i ω t ) ] ,
e 2 x = - i K [ I 0 ( e ) + I 2 ( e ) cos ( 2 θ p ) ] , e 2 y = - i K I 2 ( e ) sin ( 2 θ p ) , e 2 z = - 2 K I 1 ( e ) cos θ p ,
K = k 1 f l 0 2 = π n 1 f l 0 λ
I 0 ( e ) = 0 α ( cos ϕ 1 ) 1 / 2 sin ϕ 1 exp [ i k 0 Ψ ( ϕ 1 , ϕ 2 , - d ) ] × ( τ s + τ p cos ϕ 2 ) J 0 ( v sin ϕ 1 sin α ) exp ( i u cos ϕ 2 sin 2 α ) d ϕ 1 , I 1 ( e ) = 0 α ( cos ϕ 1 ) 1 / 2 sin ϕ 1 exp [ i k 0 Ψ ( ϕ 1 , ϕ 2 , - d ) ] × τ p sin ϕ 2 J 1 ( v sin ϕ 1 sin α ) exp ( i u cos ϕ 2 sin 2 α ) d ϕ 1 , I 2 ( e ) = 0 α ( cos ϕ 1 ) 1 / 2 sin ϕ 1 exp [ i k 0 Ψ ( ϕ 1 , ϕ 2 , - d ) ] × ( τ s - τ p cos ϕ 2 ) J 2 ( v sin ϕ 1 sin α ) exp ( i u cos ϕ 2 sin 2 α ) d ϕ 1 ,
Ψ ( ϕ 1 , ϕ 2 , - d ) = - d ( n 1 cos ϕ 1 - n 2 cos ϕ 2 ) ,
v = k 1 ( x 2 + y 2 ) 1 / 2 sin α = k 1 r p sin ϕ p sin α , u = k 2 z sin 2 α = k 2 r p cos ϕ p sin 2 α ,
I l T ( u 2 , v 2 ) = n 2 n 1 0 β F l ( cos ϕ 2 ) exp [ i k 0 Ψ ( cos ϕ 2 ) ] × J l ( v 2 sin ϕ 2 sin β ) exp ( i u 2 cos ϕ 2 sin 2 β ) d ϕ 2 ,
v 2 = k 2 ( x 2 + y 2 ) 1 / 2 sin β = k 2 r p sin ϕ p sin β , u 2 = k 2 z sin 2 β = k 2 r p cos ϕ p sin 2 β ,

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